LFM Pure

jee-main 2022 Q66 Area and Geometric Measurement Involving Circles View

Let $C$ be the centre of the circle $x^2 + y^2 - x + 2y = \frac{11}{4}$ and $P$ be a point on the circle. A line passes through the point $C$, makes an angle of $\frac{\pi}{4}$ with the line $CP$ and intersects the circle at the points $Q$ and $R$. Then the area of the triangle $PQR$ (in unit$^2$) is
(1) 2
(2) $2\sqrt{2}$
(3) $8\sin\frac{\pi}{8}$
(4) $8\cos\frac{\pi}{8}$

jee-main 2022 Q66 Circle Identification and Classification View

The set of values of $k$ for which the circle $C : 4 x ^ { 2 } + 4 y ^ { 2 } - 12 x + 8 y + k = 0$ lies inside the fourth quadrant and the point $\left( 1 , - \frac { 1 } { 3 } \right)$ lies on or inside the circle $C$ is
(1) An empty set
(2) $\left( 6 , \frac { 95 } { 9 } \right]$
(3) $\left[ \frac { 80 } { 9 } , 10 \right)$
(4) $\left( 9 , \frac { 92 } { 9 } \right]$

jee-main 2022 Q66 Tangent Lines and Tangent Lengths View

Let $P ( a , b )$ be a point on the parabola $y ^ { 2 } = 8 x$ such that the tangent at $P$ passes through the centre of the circle $x ^ { 2 } + y ^ { 2 } - 10 x - 14 y + 65 = 0$. Let $A$ be the product of all possible values of $a$ and $B$ be the product of all possible values of $b$. Then the value of $A + B$ is equal to
(1) 0
(2) 25
(3) 40
(4) 65

jee-main 2022 Q66 Tangent Lines and Tangent Lengths View

A circle $C _ { 1 }$ passes through the origin $O$ and has diameter 4 on the positive $x$-axis. The line $y = 2 x$ gives a chord $O A$ of a circle $C _ { 1 }$. Let $C _ { 2 }$ be the circle with $O A$ as a diameter. If the tangent to $C _ { 2 }$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $Q A : A P$ is equal to
(1) $1 : 4$
(2) $1 : 5$
(3) $2 : 5$
(4) $1 : 3$

jee-main 2022 Q67 Circle-Related Locus Problems View

A circle touches both the $y$-axis and the line $x + y = 0$. Then the locus of its center is
(1) $y = \sqrt{2}x$
(2) $x = \sqrt{2}y$
(3) $y^2 - x^2 = 2xy$
(4) $x^2 - y^2 = 2xy$

jee-main 2022 Q67 Eccentricity or Asymptote Computation View

If the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ meets the line $\frac{x}{7} + \frac{y}{2\sqrt{6}} = 1$ on the $x$-axis and the line $\frac{x}{7} - \frac{y}{2\sqrt{6}} = 1$ on the $y$-axis, then the eccentricity of the ellipse is
(1) $\frac{5}{7}$
(2) $\frac{2\sqrt{6}}{7}$
(3) $\frac{3}{7}$
(4) $\frac{2\sqrt{5}}{7}$

jee-main 2022 Q67 Triangle or Quadrilateral Area and Perimeter with Foci View

If the tangents drawn at the points $P$ and $Q$ on the parabola $y^2 = 2x - 3$ intersect at the point $R(0, 1)$, then the orthocentre of the triangle $PQR$ is
(1) $(0, 1)$
(2) $(2, -1)$
(3) $(6, 3)$
(4) $(2, 1)$

jee-main 2022 Q67 Inscribed/Circumscribed Circle Computations View

Let $A(\alpha, -2)$, $B(\alpha, 6)$ and $C\left(\frac{\alpha}{4}, -2\right)$ be vertices of a $\triangle ABC$. If $\left(5, \frac{\alpha}{4}\right)$ is the circumcentre of $\triangle ABC$, then which of the following is NOT correct about $\triangle ABC$
(1) area is 24
(2) perimeter is 25
(3) circumradius is 5
(4) inradius is 2

jee-main 2022 Q67 Circle Equation Derivation View

If the length of the latus rectum of a parabola, whose focus is $( a , a )$ and the tangent at its vertex is $x + y = a$, is 16 , then $| a |$ is equal to
(1) $2 \sqrt { 2 }$
(2) $2 \sqrt { 3 }$
(3) $4 \sqrt { 2 }$
(4) 4

jee-main 2022 Q68 Chord Length and Chord Properties View

The line $y = x + 1$ meets the ellipse $\frac{x^2}{4} + \frac{y^2}{2} = 1$ at two points $P$ and $Q$. If $r$ is the radius of the circle with $PQ$ as diameter then $3r^2$ is equal to
(1) 20
(2) 12
(3) 11
(4) 8

jee-main 2022 Q68 Eccentricity or Asymptote Computation View

Let the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{7} = 1$ and the hyperbola $\frac{x^2}{144} - \frac{y^2}{\alpha} = \frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is:
(1) $\frac{32}{9}$
(2) $\frac{18}{5}$
(3) $\frac{27}{4}$
(4) $\frac{27}{10}$

jee-main 2022 Q83 Circle Equation Derivation View

If the length of the latus rectum of the ellipse $x ^ { 2 } + 4 y ^ { 2 } + 2 x + 8 y - \lambda = 0$ is 4 , and $l$ is the length of its major axis, then $\lambda + l$ is equal to $\_\_\_\_$ .

jee-main 2022 Q83 Tangent and Normal Line Problems View

A common tangent $T$ to the curves $C _ { 1 } : \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } = 1$ and $C _ { 2 } : \frac { x ^ { 2 } } { 42 } - \frac { y ^ { 2 } } { 143 } = 1$ does not pass through the fourth quadrant. If $T$ touches $C _ { 1 }$ at $\left( x _ { 1 } , y _ { 1 } \right)$ and $C _ { 2 }$ at $\left( x _ { 2 } , y _ { 2 } \right)$, then $\left| 2 x _ { 1 } + x _ { 2 } \right|$ is equal to $\_\_\_\_$ .

jee-main 2022 Q84 Chord Length and Chord Properties View

Let a circle $C : ( x - h ) ^ { 2 } + ( y - k ) ^ { 2 } = r ^ { 2 } , k > 0$, touch the $x$-axis at $( 1,0 )$. If the line $x + y = 0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $P Q$ is 2, then the value of $h + k + r$ is equal to $\_\_\_\_$.

jee-main 2022 Q84 Equation Determination from Geometric Conditions View

An ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ passes through the vertices of the hyperbola $H : \frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 64 } = - 1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac { 1 } { 2 }$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113 l$ is equal to $\_\_\_\_$ .

jee-main 2022 Q85 Tangent and Normal Line Problems View

Let the common tangents to the curves $4 \left( x ^ { 2 } + y ^ { 2 } \right) = 9$ and $y ^ { 2 } = 4 x$ intersect at the point $Q$. Let an ellipse, centered at the origin $O$, has lengths of semi-minor and semi-major axes equal to $OQ$ and 6, respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\frac { l } { e ^ { 2 } }$ is equal to $\_\_\_\_$.

jee-main 2022 Q85 Circles Tangent to Each Other or to Axes View

The sum of diameters of the circles that touch (i) the parabola $75x ^ { 2 } = 64(5y - 3)$ at the point $\left(\frac { 8 } { 5 }, \frac { 6 } { 5 }\right)$ and (ii) the $y$-axis, is equal to $\_\_\_\_$.

jee-main 2022 Q87 Tangent Lines and Tangent Lengths View

Two tangent lines $l _ { 1 }$ and $l _ { 2 }$ are drawn from the point $( 2,0 )$ to the parabola $2 y ^ { 2 } = - x$. If the lines $l _ { 1 }$ and $l _ { 2 }$ are also tangent to the circle $( x - 5 ) ^ { 2 } + y ^ { 2 } = r$, then $17 r ^ { 2 }$ is equal to $\_\_\_\_$.

jee-main 2022 Q88 Tangent Lines and Tangent Lengths View

Let the tangents at the points $P$ and $Q$ on the ellipse $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 4 } = 1$ meet at the point $R ( \sqrt { 2 } , 2 \sqrt { 2 } - 2 )$. If $S$ is the focus of the ellipse on its negative major axis, then $S P ^ { 2 } + S Q ^ { 2 }$ is equal to $\_\_\_\_$.

jee-main 2023 Q64 Locus and Trajectory Derivation View

Let a tangent to the curve $y ^ { 2 } = 24 x$ meet the curve $x y = 2$ at the points $A$ and $B$. Then the midpoints of such line segments $A B$ lie on a parabola with the
(1) directrix $4 x = 3$
(2) directrix $4 x = - 3$
(3) Length of latus rectum $\frac { 3 } { 2 }$
(4) Length of latus rectum 2

jee-main 2023 Q64 Area and Geometric Measurement Involving Circles View

Let a circle $C_1$ be obtained on rolling the circle $x^2 + y^2 - 4x - 6y + 11 = 0$ upwards 4 units on the tangent $T$ to it at the point $(3,2)$. Let $C_2$ be the image of $C_1$ in $T$. Let $A$ and $B$ be the centers of circles $C_1$ and $C_2$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$-axis. Then the area of the trapezium AMNB is:
(1) $22 + \sqrt{2}$
(2) $41 + \sqrt{2}$
(3) $3 + 2\sqrt{2}$
(4) $21 + \sqrt{2}$

jee-main 2023 Q65 Eccentricity or Asymptote Computation View

If the maximum distance of normal to the ellipse $\frac{x^2}{4} + \frac{y^2}{b^2} = 1$, $b < 2$, from the origin is 1, then the eccentricity of the ellipse is:
(1) $\frac{1}{\sqrt{2}}$
(2) $\frac{\sqrt{3}}{2}$
(3) $\frac{1}{2}$
(4) $\frac{\sqrt{3}}{4}$

jee-main 2023 Q65 Circle-Related Locus Problems View

A line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment $AB$ in the ratio $2:3$, is a circle of radius
(1) $\frac{3}{5}\lambda$
(2) $\frac{2}{3}\lambda$
(3) $\frac{\sqrt{19}}{5}\lambda$
(4) $\frac{\sqrt{19}}{7}\lambda$

jee-main 2023 Q66 Area and Geometric Measurement Involving Circles View

Let the ellipse $E$: $x^2 + 9y^2 = 9$ intersect the positive $x$- and $y$-axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle with vertices $A$, $P$ and the origin $O$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m - n$ is equal to
(1) 16
(2) 15
(3) 17
(4) 18

jee-main 2023 Q67 Circle-Related Locus Problems View

Let $A$ be the point $(1, 2)$ and $B$ be any point on the curve $x ^ { 2 } + y ^ { 2 } = 16$. If the centre of the locus of the point $P$, which divides the line segment $AB$ in the ratio $3 : 2$ is the point $C( \alpha , \beta )$, then the length of the line segment $AC$ is
(1) $\frac { 3 \sqrt { 5 } } { 5 }$
(2) $\frac { 4 \sqrt { 5 } } { 5 }$
(3) $\frac { 2 \sqrt { 5 } } { 5 }$
(4) $\frac { 6 \sqrt { 5 } } { 5 }$

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LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

Standard trigonometric equations 142

Trig Proofs 92

Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

Linear transformations 72

Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

Small angle approximation 14

Chain Rule 281

Product & Quotient Rules 13

Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164